Chemistry of Engineering Materials PDF

# Chemistry of Engineering Materials ## M5. Basic Concepts of Crystal Structure ### 5.1 Introduction ### 5.2 Types of Crystalline Solids ### 5.3 Metallic Crystal Structures ### 5.4 Close-Packed Crystal Structures ### 5.5 X-ray Diffraction and Bragg's Law ## Learning Outcomes - Classify solids...

# Chemistry of Engineering Materials ## M5. Basic Concepts of Crystal Structure ### 5.1 Introduction ### 5.2 Types of Crystalline Solids ### 5.3 Metallic Crystal Structures ### 5.4 Close-Packed Crystal Structures ### 5.5 X-ray Diffraction and Bragg's Law ## Learning Outcomes - Classify solids based on their bonding/intermolecular forces and understand how difference in bonding relates to physical properties - Know the difference between crystalline and amorphous solids - Describe arrangement of atoms in a common cubic crystal lattice - Recognize the four two-dimensional and the seven three-dimensional primitive lattices. - Calculate the empirical formula and density of ionic and metallic solids from a picture of the unit cell - Describe the differences between substitutional alloys, interstitial alloys, and intermetallic compounds. ## Course Outcomes - Explain the chemical principles and concepts of structures and bonding of common materials ## Introduction Modern devices like computers and cell phones are built from solids with very specific physical properties. For example, the integrated circuit that is at the heart of many electronic devices is built from semiconductors like silicon, metals like copper, and insulators like hafnium oxide. Hard drives, which store information in computers and other devices, consist of a thin layer of a magnetic alloy deposited on glass substrate. Scientists and engineers turn almost exclusively to solids for materials used in many other technologies: alloys for magnets and airplane turbines, semiconductors for solar cells and light-emitting diodes, polymers for packaging and biomedical applications. ## Metals, Nonmetals and Metalloids The ionization energy of an atom or ion is the minimum energy required to remove an electron from the ground state of the isolated gaseous atom or ion. Metals tend to have low ionization energies and, therefore, can form cations relatively easily. Among the fundamental atomic properties (radius, electron configuration, electron affinity, and so forth), first ionization energy is the best indicator of whether an element behaves as a metal or a nonmetal. ### Table 7.3: Characteristic Properties of Metals and Nonmetals | | Metals | Nonmetals | |----------------------|---------------------------------------------------------------|-------------------------------------------------------------------------| | **Characteristics** | | | | **Appearance** | Have a shiny luster, various colors, although most are silvery | Do not have a luster, various colors | | **Solid state** | Solids are malleable and ductile | Solids are usually brittle; some are hard, and some are soft | | **Conductivity** | Good conductors of heat and electricity | Poor conductors of heat and electricity | | **Oxides in solution** | Most metal oxides are ionic solids that are basic | Most nonmetal oxides are molecular substances that form acidic solutions | | **Ionic form** | Tend to form cations in aqueous solution | Tend to form anions or oxyanions in aqueous solution | ## Classifications of solids according to predominant bonding type ### 1. Metallic Solids Metallic solids are held together by a delocalized "sea" of collectively shared valence electrons. This form of bonding allows metals to conduct electricity and also responsible for the fact that most metals are relatively strong without being brittle. ### 2. Ionic Solids Ionic solids are held together by the mutual attraction between cations and anions. Differences between ionic and metallic bonding make the electrical and mechanical properties of ionic solids very different from those of metals. ### 3. Covalent-network Solids Covalent-network solids are held together by an extended network of covalent bonds. This type of bonding can result in materials that are extremely hard, like diamond, and it is also responsible for the unique properties of semiconductors. ### 4. Molecular Solids Molecular solids are held together by the intermolecular forces: dispersion forces, dipole-dipole interactions, and hydrogen bonds. Because these forces are relatively weak, molecular solids tend to be soft and have low melting points. ## Structure of Solids Solids in which atoms are arranged in an orderly repeating pattern are called crystalline solids. These solids usually have flat surfaces, or faces, that make definite angles with one another. The orderly arrangements of atoms that produce these faces also cause the solids to have highly regular shapes. Examples of crystalline solids include sodium chloride, quartz, and diamond. Amorphous solids (from the Greek word for "without form") lack the order found in crystalline solids. At the atomic level the structures of amorphous solids are similar to the structures of liquids, but the molecules, atoms, and/or ions lack the freedom of motion they have in liquids. Amorphous solids do not have the well-defined faces and shapes of a crystal. Familiar amorphous solids are rubber, glass, and obsidian (volcanic glass). ## Crystalline Solids In a crystalline solid there is a relatively small repeating unit, called a unit cell, that is made up of a unique arrangement of atoms and embodies the structure of the solid. The structure of a crystalline solid is defined by (a) the size and shape of the unit cell and (b) the locations of atoms within the unit cell. The structure of the crystal can be built by stacking this unit over and over in all three dimensions. The geometrical pattern of points on which the unit cells are arranged is called a crystal lattice. The crystal lattice is, in effect, an abstract (that is, not real) scaffolding for the crystal structure. ## Two-Dimensional Lattices There are five two-dimensional lattices. ### 1. Rectangular Lattice **$a \neq b, \gamma = 90^{\circ}$** ### 2. Hexagonal Lattice **$a = b, \gamma = 120^{\circ}$** ### 3. Oblique Lattice **$a \neq b, \gamma = arbitrary$** ### 4. Square Lattice **$a = b, \gamma = 90^{\circ}$** ### 5. Rhombic Lattice **$a = b, \gamma = arbitrary$** ## The Seven Three-Dimensional Primitive Lattices **1. Cubic** $a = b = c, \alpha = \beta = \gamma = 90^{\circ}$ **2. Tetragonal** $a = b \neq c, \alpha = \beta = \gamma = 90^{\circ}$ **3. Orthorhombic** $a \neq b \neq c, \alpha = \beta = \gamma = 90^{\circ}$ **4. Rhombohedral** $a = b = c, \alpha = \beta = \gamma \neq 90^{\circ}$ **5. Hexagonal** $a = b \neq c, \alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}$ **6. Monoclinic** $a \neq b \neq c, \alpha = \gamma = 90^{\circ}, \beta \neq 90^{\circ}$ **7. Triclinic** $a\neq b \neq c, \alpha \neq \beta \neq \gamma$ ## Three-Dimensional Lattices ### 1. Body-centered cubic lattice A body-centered cubic lattice has one lattice point at the center of the unit cell in addition to the lattice points at the eight corners. ### 2. Primitive cubic lattice A lattice point at each corner of a unit cell gives a primitive lattice. ### 3. Face-centered cubic lattice A face-centered cubic lattice has one lattice point at the center of each of the six faces of the unit cell in addition to the lattice points at the eight corners. ## Filling the Unit Cell The lattice by itself does not define a crystal structure. In the simplest case, the crystal structure consists of identical atoms, and each atom lies directly on a lattice point. When this happens, the crystal structure and the lattice points have identical patterns. Many metallic elements adopt such structures. In most crystals the atoms are not exactly coincident with the lattice points. Instead, a group of atoms, called a motif, is associated with each lattice point. The unit cell contains a specific motif of atoms, and the crystal structure is built up by repeating the unit cell over and over. ## Unit Cell and Crystal Lattice Two-dimensional structure of graphene built up from a single unit cell. ## Structure of Metallic Solids The crystal structures of most metals are simple enough that we can generate the structure by placing a single atom on each lattice point. Metals with a primitive cubic structure are rare, one of the few examples being the radioactive element polonium. Body-centered cubic metals include iron, chromium, sodium, and tungsten. Examples of face-centered cubic metals include aluminum, lead, copper, silver, and gold. ## Fraction of Any Atom as a Function of Location Within the Unit Cell | Atom Location | Number of Unit Cells Sharing Atom | Fraction of Atom Within Unit Cell | |----------------|-------------------------------------|-------------------------------------| | Corner | 8 | 1/8 or 12.5% | | Edge | 4 | 1/4 or 25% | | Face | 2 | 1/2 or 50% | | Anywhere else | 1 | 1 or 100% | ## Determination of no. of atoms within a unit cell What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice? **Solution** * At the corners: 1/8 x 8 = 1 atom * At the faces: 1/2 x 6 = 3 atoms * Total atoms within the unit cell = 4 atoms ## Close Packing The shortage of valence electrons and the fact that they are collectively shared make it favorable for the atoms in a metal to pack together closely. Because atoms are are spherical objects, we can understand the structures of metals by considering how spheres pack. The most efficient way to pack one layer of equal-sized spheres is to surround each sphere by six neighbors. ## Hexagonal close packing vs. Cubic close packing ### Hexagonal Close Packing (hcp) * Isometric view showing depression * Second layer top view * Third layer top view ### Cubic Close Packing (ccp) * First layer top view showing depression ## hcp vs. ccp In both hexagonal close packing and cubic close packing, each sphere has 12 equidistant nearest neighbors: six neighbors in the same layer, three from the layer above, and three from the layer below. Each sphere has a coordination number of 12. The coordination number is the number of atoms immediately surrounding a given atom in a crystal structure. The unit cells for (a) a hexagonal close-packed metal and (b) a cubic close-packed metal. The solid lines indicate the unit cell boundaries. ## Unit cell: ccp By rotating our perspective, we can see that a CCP structure has a unit cell with a face containing an atom from layer A at one corner, atoms from layer B across a diagonal (at two corners and in the middle of the face), and an atom from layer C at the remaining corner. This is the same as a face-centered cubic arrangement. ## Ionic Solids Ionic solids are held together by the electrostatic attraction between cations and anions-ionic bonds. The high melting and boiling points of ionic compounds are a testament to the strength of the ionic bonds. The strength of an ionic bond depends on the charges and sizes of the ions. The attractions between cations and anions increase as the charges of the ions go up. Thus NaCl, where the ions have charges of 1+ and 1-, melts at 801°C, whereas MgO, where the ions have charges of 2+ and 2-, melts at 2852°C. The interactions between cations and anions also increase as the ions get smaller, as we see from the melting points of the alkali metal halides. Although ionic and metallic solids both have high melting and boiling points, the differences between ionic and metallic bonding are responsible for important differences in their properties..Because the valence electrons in ionic compounds are confined to the anions, rather than being delocalized, ionic compounds are typically electrical insulators. They tend to be brittle, a property explained by repulsive interactions between ions of like charge. Brittleness is not necessarily a negative quality, of course, as evidenced by the beauty of a cut gemstone. The multiple facets that are the basis of this beauty are possible because crystals cleave along well-defined directions with respect to the crystalline lattice. ## Three common ionic structure types The structures of CsCl, NaCl, and ZnS. Each structure type can be generated by the combination of a two-atom motif and the appropriate lattice. ## Coordination environments in CsCl, NaCl, and ZnS. The sizes of the ions have been reduced to show the coordination environments clearly. Remember that, in ionic crystals, ions of opposite charge touch each other but ions of the same charge should not touch. ## Sample Problem 5.1 What is the total number of atoms of each kind within the unit cell shown? * **Solution** * No. of Fe atoms: 1/8 x 8 = 1 atom * No. of Sr atoms: 1 x 1 = 1 atom * No. of O atoms: 1/4 x 8 = 2 atoms ## Sample Problem 5.2 The unit cell of a binary compound of copper and oxygen is shown here. Given this image and the ionic radii **rcu+ = 0.74 Å** and **ro2- = 1.26 Å**, * **(a) determine the empirical formula of this compound** * No. Cu+ ions: 1x1 = 1 * No. O2- ions: 1/8 x 8 = 1 * **Empirical Formula: Cu₂O** * **(b) Determine coordination numbers of copper and oxygen** * **For Copper** * Cation coordination number: 2 * **For Oxygen** * Number of cations per formula unit: 2 * Number of anions per formula unit: 1 * Anion coordination number = (cation coordination number x number of cations per formula unit) / number of anions per formula unit * =2 x 2/1 = 4 * **(c) Estimate the length of the edge of the cubic unit cell** * y = 4(ro2-) + 4(rcu+) = 4(1.26 Å) + 4(0.74 Å) * y = 8.00 Å = 8.00 x10^-8 cm * a = √3/8 x 8.00 x 10^-8 cm = 4.62 x 10^-8 cm * **(d) Estimate the density of the compound** * m = 4(65.55 amu) + 2(16.00 amu) = 294.2 amu = 4.885 x 10^-22 g * V = a³ = (4.62 x 10^-8 cm)³ = 9.85 x 10^-23 cm³ * p = m/V = 4.885 x 10^-22 g / 9.85 x 10^-23 cm³ = 4.96 g/cm³ ## Diffraction When light waves pass through a narrow slit, they are scattered in such a way that the wave seems to spread out. This physical phenomenon is called diffraction. When light passes through many evenly spaced narrow slits (a diffraction grating), the scattered waves interact to form a series of bright and dark bands, known as a diffraction pattern. The pattern consists of a central bright fringe flanked by much weaker maxima alternating with dark fringes. ## Single-Slit Diffraction A single-slit diffraction experiment. ## Diffraction from a Single Slit The diffraction pattern consists of a central bright band, which may be much broader than the width of the slit, bordered by alternating dark and bright bands with rapidly decreasing intensity. ## X-Ray Diffraction X rays were discovered by Wilhelm Röntgen (1845-1923) in 1895, and early experiments suggested that they were electromagnetic waves with wavelengths of the order of 0.1 nm. At about the same time, the idea began to emerge that in a crystalline solid the atoms are arranged in a regular repeating pattern, with spacing between adjacent atoms also of the order of Putting these two ideas together, Max von Laue (1879-1960) proposed in 1912 that a crystal might serve as a kind of three-dimensional diffraction grating for x rays. That is, a beam of x rays might be scattered (that is, absorbed and re-emitted) by the individual atoms in a crystal, and the scattered waves might interfere just like waves from a diffraction grating. ## Diffraction from a Single Slit The first x-ray diffraction experiments were performed in 1912 by Friederich, Knipping, and von Laue, using the experimental setup shown below. The scattered x rays did form an interference pattern, which they recorded on photographic film. ## X-Ray Diffraction These experiments verified that x rays are waves, or at least have wavelike properties, and also that the atoms in a crystal are arranged in a regular pattern. Since that time, x-ray diffraction has proved to be an invaluable research tool, both for measuring x-ray wavelengths and for studying the structure of crystals and complex molecules. The spacing of the layers of atoms in solid crystals is usually about 2-20 Å. The wavelengths of X-rays are also in this range. Thus, a crystal can serve as an effective diffraction grating for X-rays. X-ray diffraction results from the scattering of X-rays by a regular arrangement of atoms, molecules, or ions. Much of what we know about crystal structures has been obtained by looking at the diffraction patterns that result when X-rays pass through a crystal, a technique known as X-ray crystallography. ## X-Ray Diffraction X-ray crystallography is used extensively to determine the structures of molecules in crystals. The instruments used to measure X-ray diffraction, known as X-ray diffractometers, are now computer controlled, making the collection of diffraction data highly automated. The diffraction pattern of a crystal can be determined very accurately and quickly (sometimes in a matter of hours), even though thousands of diffraction spots are measured. Computer programs are then used to analyze the diffraction data and determine the arrangement and structure of the molecules in the crystal. X-ray diffraction is an important technique in industries ranging from steel and cement manufacture to pharmaceuticals. ## X-Ray Diffraction and Bragg's Law The arrangement of atoms in a crystal of sodium chloride (NaCl) is shown in Figure 38.22. Each unit cell (the geometric solid that repeats throughout the crystal) is a cube having an edge length a. A careful examination of the NaCl structure shows that the ions lie in discrete planes (the shaded areas in Fig. 38.22). The same is true for reflection from the entire family of parallel planes. Hence, the condition for constructive interference (maxima in the reflected beam) is **2d sin 0 = ηλ** n=1, 2, 3, This condition is known as **Bragg's law**, after W. L. Bragg (1890-1971), who first derived the relationship. If the wavelength and diffraction angle are measured, equation above can be used to calculate the spacing between atomic planes. ## Diffraction Now suppose an incident x-ray beam makes an angle & with one of the planes as in Figure 38.23. The beam can be reflected from both the upper plane and the lower one, but the beam reflected from the lower plane travels farther than the beam reflected from the upper plane. The effective path difference is 2d sin 0. The two beams reinforce each other (constructive interference) when this path difference equals some integer multiple of λ.